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SDThe Riemann Zeta Function has been successfully and promisingly
generalized in various ways so that the concept of zeta functions
has become important in many different areas of research. In
particular, work done by Y. Ihara in the 1960s led to the
definition of an Ihara Zeta Function for finite graphs. The Ihara
Zeta Function has the nice property of having three equivalent
expressions: an Euler product form over ``primes" of the graph, an
expression in terms of vertex operators on the graph, and an
expression in terms of arc operators on the graph. In this paper
we present two possibilities for generalizing the Ihara Zeta
Function to cell products of graphs. We start with a background
discussion of the Ihara Zeta Function and cell products. Then we
present our generalized zeta functions and prove some properties
about them. Our hope is that the ideas presented in this paper
will stimulate further ideas about using the nice properties of
the Ihara Zeta Function as a model for defining zeta functions
more generally on higher dimensional geometric objects.
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